In my studies, I have come across an interesting problem that I need some help with. I'll get right to it.
Suppose that we have some function $$f(x,y)= \mathbb{E}[(x+yZ)\mathbf{1}_{|x+yZ|<1}]$$ where $Z$ is a standard normal. For the standard Brownian Motion $(W_t)_{t\in[0,1]}$, the random variable $X$ is $$X= B_1 \mathbf{1}_{|B_1|\ge1} - B_1 \mathbf{1}_{|B_1|<1}$$
and I let $(X_t)_{t\in[0,1]}$ be defined as $X_t = \mathbb{E}[X|\mathcal{F}_t]$. The filtration is that which is generated by $B$ up to $t$.
It is clear to me that $X$ is a standard normal random variable. It is also clear that $X_t$ is a martingale. What I have difficulties showing is the following equivalence: $$X_t = B_t - 2f(B_t, \sqrt{1-t})$$
I have tried using the equivalence of distribution of $B_t+\sqrt{1-t}Z$ to $B_1$, but I was not having much luck in my workings thereafter.
Any help is much appreciated!